Donaldson's ground breaking result says that the study of symplectic 4-manifolds up to blow-up is equivalent to that of Lefschetz fibrations over the 2-sphere. This relationship has been extended to the relative case by Akbulut-Ozbagci [1]} and Loi-Piergallini [2]} who established a correspondence between Stein manifolds and positive allowable Lefschetz fibrations. Working with Lefschetz fibrations is more preferable as the topology of the total space, which is a 4-dimensional manifold, is completely determined by a monodromy factorization in mapping class groups, which can be understood through 2-dimensional techniques. It remains a good challenge for 4-dimensional topologist to compute invariants of a Lefschetz fibration out of its monodromy factorization. Signature is perhaps the simplest non-trivial invariant among them. Recall that the signature \(\sigma (Y)\)  of a compact oriented 4-manifold \(Y\)  is the signature of the intersection form on the second homology group \(H_2(Y;{\mathbb {Z}}).\)
